Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | D4×#217 |
| Sylow | C3 ⋊ D4×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting, prime radical | C23×C3×D4 | C2 |
| socle, cosocle | C23×C3 | C24 |
| hypercenter | C23×D4 | D3 |
| hypocenter | C3 | D4×C22⋊C4 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×D4 | |
1252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(13,15);(2,3)(4,5,6,7)(12,13)(14,15);(9,11);(8,9)(10,11)90 conjugacy classes — view the full character table.
| maximal subgroup of | #10691604, #10691608, #10691632, … |
| maximal normal subgroup of | #10691604, #10691608, #10691632, … |
| maximal quotient of | C22⋊C4×#217, C4⋊C4×#217, D4×C3⋊#95, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16621);